3.537 \(\int (d+e x) \left (a+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=107 \[ \frac{5 a^3 d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 \sqrt{c}}+\frac{5}{16} a^2 d x \sqrt{a+c x^2}+\frac{1}{6} d x \left (a+c x^2\right )^{5/2}+\frac{5}{24} a d x \left (a+c x^2\right )^{3/2}+\frac{e \left (a+c x^2\right )^{7/2}}{7 c} \]

[Out]

(5*a^2*d*x*Sqrt[a + c*x^2])/16 + (5*a*d*x*(a + c*x^2)^(3/2))/24 + (d*x*(a + c*x^
2)^(5/2))/6 + (e*(a + c*x^2)^(7/2))/(7*c) + (5*a^3*d*ArcTanh[(Sqrt[c]*x)/Sqrt[a
+ c*x^2]])/(16*Sqrt[c])

_______________________________________________________________________________________

Rubi [A]  time = 0.0999796, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{5 a^3 d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 \sqrt{c}}+\frac{5}{16} a^2 d x \sqrt{a+c x^2}+\frac{1}{6} d x \left (a+c x^2\right )^{5/2}+\frac{5}{24} a d x \left (a+c x^2\right )^{3/2}+\frac{e \left (a+c x^2\right )^{7/2}}{7 c} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)*(a + c*x^2)^(5/2),x]

[Out]

(5*a^2*d*x*Sqrt[a + c*x^2])/16 + (5*a*d*x*(a + c*x^2)^(3/2))/24 + (d*x*(a + c*x^
2)^(5/2))/6 + (e*(a + c*x^2)^(7/2))/(7*c) + (5*a^3*d*ArcTanh[(Sqrt[c]*x)/Sqrt[a
+ c*x^2]])/(16*Sqrt[c])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.2141, size = 100, normalized size = 0.93 \[ \frac{5 a^{3} d \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{16 \sqrt{c}} + \frac{5 a^{2} d x \sqrt{a + c x^{2}}}{16} + \frac{5 a d x \left (a + c x^{2}\right )^{\frac{3}{2}}}{24} + \frac{d x \left (a + c x^{2}\right )^{\frac{5}{2}}}{6} + \frac{e \left (a + c x^{2}\right )^{\frac{7}{2}}}{7 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)*(c*x**2+a)**(5/2),x)

[Out]

5*a**3*d*atanh(sqrt(c)*x/sqrt(a + c*x**2))/(16*sqrt(c)) + 5*a**2*d*x*sqrt(a + c*
x**2)/16 + 5*a*d*x*(a + c*x**2)**(3/2)/24 + d*x*(a + c*x**2)**(5/2)/6 + e*(a + c
*x**2)**(7/2)/(7*c)

_______________________________________________________________________________________

Mathematica [A]  time = 0.115728, size = 108, normalized size = 1.01 \[ \frac{105 a^3 \sqrt{c} d \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )+\sqrt{a+c x^2} \left (48 a^3 e+3 a^2 c x (77 d+48 e x)+2 a c^2 x^3 (91 d+72 e x)+8 c^3 x^5 (7 d+6 e x)\right )}{336 c} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)*(a + c*x^2)^(5/2),x]

[Out]

(Sqrt[a + c*x^2]*(48*a^3*e + 8*c^3*x^5*(7*d + 6*e*x) + 3*a^2*c*x*(77*d + 48*e*x)
+ 2*a*c^2*x^3*(91*d + 72*e*x)) + 105*a^3*Sqrt[c]*d*Log[c*x + Sqrt[c]*Sqrt[a + c
*x^2]])/(336*c)

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 85, normalized size = 0.8 \[{\frac{dx}{6} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,adx}{24} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}dx}{16}\sqrt{c{x}^{2}+a}}+{\frac{5\,{a}^{3}d}{16}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{e}{7\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(c*x^2+a)^(5/2),x)

[Out]

1/6*d*x*(c*x^2+a)^(5/2)+5/24*a*d*x*(c*x^2+a)^(3/2)+5/16*a^2*d*x*(c*x^2+a)^(1/2)+
5/16*d*a^3/c^(1/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2))+1/7*e*(c*x^2+a)^(7/2)/c

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)*(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.24574, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, a^{3} c d \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right ) + 2 \,{\left (48 \, c^{3} e x^{6} + 56 \, c^{3} d x^{5} + 144 \, a c^{2} e x^{4} + 182 \, a c^{2} d x^{3} + 144 \, a^{2} c e x^{2} + 231 \, a^{2} c d x + 48 \, a^{3} e\right )} \sqrt{c x^{2} + a} \sqrt{c}}{672 \, c^{\frac{3}{2}}}, \frac{105 \, a^{3} c d \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) +{\left (48 \, c^{3} e x^{6} + 56 \, c^{3} d x^{5} + 144 \, a c^{2} e x^{4} + 182 \, a c^{2} d x^{3} + 144 \, a^{2} c e x^{2} + 231 \, a^{2} c d x + 48 \, a^{3} e\right )} \sqrt{c x^{2} + a} \sqrt{-c}}{336 \, \sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)*(e*x + d),x, algorithm="fricas")

[Out]

[1/672*(105*a^3*c*d*log(-2*sqrt(c*x^2 + a)*c*x - (2*c*x^2 + a)*sqrt(c)) + 2*(48*
c^3*e*x^6 + 56*c^3*d*x^5 + 144*a*c^2*e*x^4 + 182*a*c^2*d*x^3 + 144*a^2*c*e*x^2 +
231*a^2*c*d*x + 48*a^3*e)*sqrt(c*x^2 + a)*sqrt(c))/c^(3/2), 1/336*(105*a^3*c*d*
arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + (48*c^3*e*x^6 + 56*c^3*d*x^5 + 144*a*c^2*e*
x^4 + 182*a*c^2*d*x^3 + 144*a^2*c*e*x^2 + 231*a^2*c*d*x + 48*a^3*e)*sqrt(c*x^2 +
a)*sqrt(-c))/(sqrt(-c)*c)]

_______________________________________________________________________________________

Sympy [A]  time = 38.0144, size = 348, normalized size = 3.25 \[ \frac{a^{\frac{5}{2}} d x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{3 a^{\frac{5}{2}} d x}{16 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{35 a^{\frac{3}{2}} c d x^{3}}{48 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{17 \sqrt{a} c^{2} d x^{5}}{24 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{5 a^{3} d \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{16 \sqrt{c}} + a^{2} e \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) + 2 a c e \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + c x^{2}}}{15 c^{2}} + \frac{a x^{2} \sqrt{a + c x^{2}}}{15 c} + \frac{x^{4} \sqrt{a + c x^{2}}}{5} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) + c^{2} e \left (\begin{cases} \frac{8 a^{3} \sqrt{a + c x^{2}}}{105 c^{3}} - \frac{4 a^{2} x^{2} \sqrt{a + c x^{2}}}{105 c^{2}} + \frac{a x^{4} \sqrt{a + c x^{2}}}{35 c} + \frac{x^{6} \sqrt{a + c x^{2}}}{7} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{6}}{6} & \text{otherwise} \end{cases}\right ) + \frac{c^{3} d x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)*(c*x**2+a)**(5/2),x)

[Out]

a**(5/2)*d*x*sqrt(1 + c*x**2/a)/2 + 3*a**(5/2)*d*x/(16*sqrt(1 + c*x**2/a)) + 35*
a**(3/2)*c*d*x**3/(48*sqrt(1 + c*x**2/a)) + 17*sqrt(a)*c**2*d*x**5/(24*sqrt(1 +
c*x**2/a)) + 5*a**3*d*asinh(sqrt(c)*x/sqrt(a))/(16*sqrt(c)) + a**2*e*Piecewise((
sqrt(a)*x**2/2, Eq(c, 0)), ((a + c*x**2)**(3/2)/(3*c), True)) + 2*a*c*e*Piecewis
e((-2*a**2*sqrt(a + c*x**2)/(15*c**2) + a*x**2*sqrt(a + c*x**2)/(15*c) + x**4*sq
rt(a + c*x**2)/5, Ne(c, 0)), (sqrt(a)*x**4/4, True)) + c**2*e*Piecewise((8*a**3*
sqrt(a + c*x**2)/(105*c**3) - 4*a**2*x**2*sqrt(a + c*x**2)/(105*c**2) + a*x**4*s
qrt(a + c*x**2)/(35*c) + x**6*sqrt(a + c*x**2)/7, Ne(c, 0)), (sqrt(a)*x**6/6, Tr
ue)) + c**3*d*x**7/(6*sqrt(a)*sqrt(1 + c*x**2/a))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.216767, size = 142, normalized size = 1.33 \[ -\frac{5 \, a^{3} d{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{16 \, \sqrt{c}} + \frac{1}{336} \, \sqrt{c x^{2} + a}{\left (\frac{48 \, a^{3} e}{c} +{\left (231 \, a^{2} d + 2 \,{\left (72 \, a^{2} e +{\left (91 \, a c d + 4 \,{\left (18 \, a c e +{\left (6 \, c^{2} x e + 7 \, c^{2} d\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)*(e*x + d),x, algorithm="giac")

[Out]

-5/16*a^3*d*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/sqrt(c) + 1/336*sqrt(c*x^2 + a
)*(48*a^3*e/c + (231*a^2*d + 2*(72*a^2*e + (91*a*c*d + 4*(18*a*c*e + (6*c^2*x*e
+ 7*c^2*d)*x)*x)*x)*x)*x)